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Journal articles on the topic 'S-ring'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Hijriati, Na'imah, Sri Wahyuni, and Indah Emilia Wijayanti. "Generalization of Schur's Lemma in Ring Representations on Modules over a Commutative Ring." European Journal of Pure and Applied Mathematics 11, no. 3 (July 31, 2018): 751–61. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3285.

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Let $ R, S $ be rings with unity, $ M $ a module over $ S $, where $ S $ a commutative ring, and $ f \colon R \rightarrow S $ a ring homomorphism. A ring representation of $ R $ on $ M $ via $ f $ is a ring homomorphism $ \mu \colon R \rightarrow End_S(M) $, where $ End_S(M) $ is a ring of all $ S $-module homomorphisms on $ M $. One of the important properties in representation of rings is the Schur's Lemma. The main result of this paper is partly the generalization of Schur's Lemma in representations of rings on modules over a commutative ring

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2

Sanghare, Mamadou. "Subrings of I-rings and S-rings." International Journal of Mathematics and Mathematical Sciences 20, no. 4 (1997): 825–27. http://dx.doi.org/10.1155/s0161171297001130.

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LetRbe a non-commutative associative ring with unity1≠0, a leftR-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism ofMis an automorphism ofM. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ringRis called a left I-ring (resp. S-ring) if every leftR-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subringBof a left I-ring (resp. S-ring)Ris not in general a left I-ring (resp. S-ring) even ifRis a finitely generatedB-module, for example the ringM3(K)of3×3matrices over a fieldKis a left I-ring (resp. S-ring), whereas its subringB={[α00βα0γ0α]/α,β,γ∈K}which is a commutative ring with a non-principal Jacobson radicalJ=K.[000100000]+K.[000000100]is not an I-ring (resp. S-ring) (see [4], theorem 8). We recall that commutative I-rings (resp S-tings) are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and S-tings have been studied in [2] and [3]. A ringRis of finite representation type if it is left and right Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative rings or finite-dimensional algebras over an algebraically closed field, the classes of left I-rings, left S-rings and rings of finite representation type are identical (see [1] and [4]). A ringRis said to be a ring with polynomial identity (P. I-ring) if there exists a polynomialf(X1,X2,…,Xn),n≥2, in the non-commuting indeterminatesX1,X2,…,Xn, over the centerZofRsuch that one of the monomials offof highest total degree has coefficient1, andf(a1,a2,…,an)=0for alla1,a2,…,aninR. Throughout this paper all rings considered are associative rings with unity, and by a moduleMover a ringRwe always understand a unitary leftR-module. We useMRto emphasize thatMis a unitary rightR-module.

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3

Kwon, Min Jae, and Jung Wook Lim. "On Nonnil-S-Noetherian Rings." Mathematics 8, no. 9 (August 26, 2020): 1428. http://dx.doi.org/10.3390/math8091428.

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Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.

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4

Singh, Amit Bhooshan. "TRIANGULAR MATRIX REPRESENTATION OF SKEW GENERALIZED POWER SERIES RINGS." Asian-European Journal of Mathematics 05, no. 04 (December 2012): 1250027. http://dx.doi.org/10.1142/s1793557112500271.

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Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this paper, we study the triangular matrix representation of skew generalized power series ring R[[S, ω]] which is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev–Neumann rings and generalized power series rings. We investigate that if R is S-compatible and (S, ω)-Armendariz, then the skew generalized power series ring has same triangulating dimension as R. Furthermore, if R is a PWP ring, then skew generalized power series is also PWP ring.

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5

ZHAO, RENYU. "LEFT APP-RINGS OF SKEW GENERALIZED POWER SERIES." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 891–900. http://dx.doi.org/10.1142/s0219498811005014.

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A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any a ∈ R. Let R be a ring, (S, ≤) be a commutative strictly ordered monoid and ω: S → End (R) be a monoid homomorphism. The skew generalized power series ring [[RS, ≤, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings and Malcev–Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring [[RS, ≤, ω]]. It is shown that if (S, ≤) is a commutative strictly totally ordered monoid, ω: S→ Aut (R) a monoid homomorphism and R a ring satisfying the descending chain condition on right annihilators, then [[RS, ≤, ω]] is left APP if and only if for any S-indexed subset A of R, the ideal lR(∑a ∈ A ∑s ∈ S Rωs (a)) is right s-unital.

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6

Kim, Dong Kyu, and Jung Wook Lim. "A Note on Weakly S-Noetherian Rings." Symmetry 12, no. 3 (March 5, 2020): 419. http://dx.doi.org/10.3390/sym12030419.

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Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We call the ring R to be a weakly S-Noetherian ring if every S-finite proper ideal of R is an S-Noetherian R-module. In this article, we study some properties of weakly S-Noetherian rings. In particular, we give some conditions for the Nagata’s idealization and the amalgamated algebra to be weakly S-Noetherian rings.

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7

Patty, Henry W. "HASIL KALI LANGSUNG S-NEAR-RING DAN S-NEAR-RING BEBAS." BAREKENG: Jurnal Ilmu Matematika dan Terapan 8, no. 2 (December 1, 2014): 1–7. http://dx.doi.org/10.30598/barekengvol8iss2pp1-7.

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Hasil kali langsung near-ring Smarandache i I X Ni  dikembangkan dari hasil kali langsung near-ring dengan kondisi khusus jika paling sedikit terdapat satu anggota dari 𝑁𝑖 merupakan near ring Smarandache (S-near-ring). Sedangkan near-ring Smarandache bebas didefinisikan dengan bantuan homomorfisma near-ring Smarandache.

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8

Mazurek, Ryszard. "Left principally quasi-Baer and left APP-rings of skew generalized power series." Journal of Algebra and Its Applications 14, no. 03 (November 7, 2014): 1550038. http://dx.doi.org/10.1142/s0219498815500383.

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Let R be a ring, S a strictly ordered monoid, and ω : S → End (R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, and (skew) monoid rings. We characterize when a skew generalized power series ring R[[S, ω]] is left principally quasi-Baer and under various finiteness conditions on R we characterize when the ring R[[S, ω]] is left APP. As immediate corollaries we obtain characterizations for all aforementioned classical ring constructions to be left principally quasi-Baer or left APP. Such a general approach not only gives new results for several constructions simultaneously, but also serves the unification of already known results.

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9

Mazurek, Ryszard, and Michał Ziembowski. "On semilocal, Bézout and distributive generalized power series rings." International Journal of Algebra and Computation 25, no. 05 (August 2015): 725–44. http://dx.doi.org/10.1142/s0218196715500174.

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Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

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10

Mazurek, Ryszard. "Rota–Baxter operators on skew generalized power series rings." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450048. http://dx.doi.org/10.1142/s0219498814500480.

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Let R be a ring, S a strictly ordered monoid, and ω : S → End (R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and generalized power series rings. We characterize those subsets T of S for which the cut-off operator with respect to T is a Rota–Baxter operator on the ring R[[S, ω]]. The obtained results provide a large class of noncommutative Rota–Baxter algebras.

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11

Birkenmeier, Gary F., Jae Keol Park, and S. Tariq Rizvi. "Hulls of Ring Extensions." Canadian Mathematical Bulletin 53, no. 4 (December 1, 2010): 587–601. http://dx.doi.org/10.4153/cmb-2010-065-9.

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AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

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12

BEHBOODI, M., R. BEYRANVAND, and H. KHABAZIAN. "STRONG ZERO-DIVISORS OF NON-COMMUTATIVE RINGS." Journal of Algebra and Its Applications 08, no. 04 (August 2009): 565–80. http://dx.doi.org/10.1142/s0219498809003540.

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We introduce the set S(R) of "strong zero-divisors" in a ring R and prove that: if S(R) is finite, then R is either finite or a prime ring. When certain sets of ideals have ACC or DCC, we show that either S(R) = R or S(R) is a union of prime ideals each of which is a left or a right annihilator of a cyclic ideal. This is a finite union when R is a Noetherian ring. For a ring R with |S(R)| = p, a prime number, we characterize R for S(R) to be an ideal. Moreover R is completely characterized when R is a ring with identity and S(R) is an ideal with p2 elements. We then consider rings R for which S(R)= Z(R), the set of zero-divisors, and determine strong zero-divisors of matrix rings over commutative rings with identity.

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13

Gurkaynak, Murat, Mustafa Cengiz, Serap Akyurek, Enis Ozyar, I. Lale Atahan, and G??lten Tekuzman. "Waldeyer???s Ring Lymphomas." American Journal of Clinical Oncology 26, no. 5 (October 2003): 437–40. http://dx.doi.org/10.1097/01.coc.0000027588.56104.15.

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14

Xiao, Guangshi, and Wenting Tong. "n-Clean Rings." Algebra Colloquium 13, no. 04 (December 2006): 599–606. http://dx.doi.org/10.1142/s1005386706000538.

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Let n be a positive integer. A ring R is called n-clean if every element of R can be written as a sum of an idempotent and n units in R. The class of n-clean rings contains clean rings and (S,n)-rings (i.e., every element is a sum of no more than n units). In this paper, we investigate some properties on n-clean rings. There exists a clean and (S,3)-ring which is not an (S,2)-ring. If R is a ring satisfying (SI), then the polynomial ring R[x] is not n-clean for any positive integer n. An example shows that for any positive integer n> 1, there exists a non n-clean ring R such that the 2× 2 matrix ring M2(R) over R is n-clean.

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15

Mazurek, Ryszard. "Archimedean domains of skew generalized power series." Forum Mathematicum 32, no. 4 (July 1, 2020): 1075–93. http://dx.doi.org/10.1515/forum-2019-0187.

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AbstractA skew generalized power series ring {R[[S,\omega,\leq]]} consists of all functions from a strictly ordered monoid {(S,\leq)} to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of this ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev–Neumann series rings, the “unskewed” versions of all of these, and generalized power series rings. In this paper, we characterize the skew generalized power series rings {R[[S,\omega,\leq]]} that are left (right) Archimedean domains in the case where the order {\leq} is total, or {\leq} is semisubtotal and the monoid S is commutative torsion-free cancellative, or {\leq} is trivial and S is totally orderable. We also answer four open questions posed by Moussavi, Padashnik and Paykan regarding the rings in the title.

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16

KOIKE, KAZUTOSHI. "MORITA DUALITY AND RING EXTENSIONS." Journal of Algebra and Its Applications 12, no. 02 (December 16, 2012): 1250160. http://dx.doi.org/10.1142/s0219498812501605.

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In this paper, we show that there exists a category equivalence between certain categories of A-rings (respectively, ring extensions of A) and B-rings (respectively, ring extensions of B), where A and B are Morita dual rings. In this category equivalence, corresponding two A-ring and B-ring are Morita dual. This is an improvement of a result of Müller, which state that if a ring A has a Morita duality induced by a bimodule BQA and R is a ring extension of A such that RA and Hom A(R, Q)A are linearly compact, then R has a Morita duality induced by the bimodule S End R( Hom A(R, Q))R with S = End R( Hom A(R, Q)). We also investigate relationships between Morita duality and finite ring extensions. Particularly, we show that if A and B are Morita dual rings with B basic, then every finite triangular (respectively, normalizing) extension R of A is Morita dual to a finite triangular (respectively, normalizing) extension S of B, and we give a result about finite centralizing free extensions, which unify a result of Mano about self-duality and a result of Fuller–Haack about semigroup rings.

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17

MARKS, GREG, RYSZARD MAZUREK, and MICHAŁ ZIEMBOWSKI. "A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS." Bulletin of the Australian Mathematical Society 81, no. 3 (February 23, 2010): 361–97. http://dx.doi.org/10.1017/s0004972709001178.

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AbstractLet R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

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18

Evdokimov, Sergei, and Ilya Ponomarenko. "Coset closure of a circulant S-ring and schurity problem." Journal of Algebra and Its Applications 15, no. 04 (February 19, 2016): 1650068. http://dx.doi.org/10.1142/s0219498816500687.

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Let [Formula: see text] be a finite group. There is a natural Galois correspondence between the permutation groups containing [Formula: see text] as a regular subgroup, and the Schur rings (S-rings) over [Formula: see text]. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group [Formula: see text] is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

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19

van Huynh, Dinh. "Some characterizations of hereditarily artinian rings." Glasgow Mathematical Journal 28, no. 1 (January 1986): 21–23. http://dx.doi.org/10.1017/s0017089500006285.

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Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

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20

Burgess, W. D., and P. N. Stewart. "The characteristic ring and the “best” way to adjoin a one." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 47, no. 3 (December 1989): 483–96. http://dx.doi.org/10.1017/s1446788700033218.

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AbstractFor any ring S we define and describe its characteristic ring, k(S). It plays the rôle of the usual characteristic even in rings whose additive structure, (S, +), is complicated. The ring k(S) is an invariant of (S, +) and also reflects certain non-additive properties of S. If R is a left faithful ring without identity element, we show how to use k(R) to embed R in a ring R1 with identity. This unital overring of R inherits many ring properties of R; for instance, if R is artinian, noetherian, semiprime Goldie, regular, biregular or a V-ring, so too is R1. In the case of regularity (or generalizations thereof), R1 satisfies a universal property with respect to the adjunction of an identity

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21

FRANCE-JACKSON, HALINA. "ON SUPERNILPOTENT RADICALS WITH THE AMITSUR PROPERTY." Bulletin of the Australian Mathematical Society 80, no. 3 (June 29, 2009): 423–29. http://dx.doi.org/10.1017/s0004972709000380.

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AbstractA radical α has the Amitsur property if α(A[x])=(α(A[x])∩A)[x] for all rings A. For rings R⊆S with the same unity, we call S a finite centralizing extension of R if there exist b1,b2,…,bt∈S such that S=b1R+b2R+⋯+btR and bir=rbi for all r∈R and i=1,2,…,t. A radical α is FCE-friendly if α(S)∩R⊆α(R) for any finite centralizing extension S of a ring R. We show that if α is a supernilpotent radical whose semisimple class contains the ring ℤ of all integers and α is FCE-friendly, then α has the Amitsur property. In this way the Amitsur property of many well-known radicals such as the prime radical, the Jacobson radical, the Brown–McCoy radical, the antisimple radical and the Behrens radical can be established. Moreover, applying this condition, we will show that the upper radical 𝒰(*k) generated by the essential cover *k of the class * of all *-rings has the Amitsur property and 𝒰(*k)(A[x])=𝒰(*k)(A)[x], where a semiprime ring R is called a *-ring if the factor ring R/I is prime radical for every nonzero ideal I of R. The importance of *-rings stems from the fact that a *-ring A is Jacobson semisimple if and only if A is a primitive ring.

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22

Sun, Ying, Hui Wang, and Wei-Guo Jia. "Crystal structure of 3,3′-diisopropyl-1,1′-(pyridine-2,6-diyl)bis[1H-imidazole-2(3H)-thione]." Acta Crystallographica Section E Crystallographic Communications 71, no. 4 (March 25, 2015): o255. http://dx.doi.org/10.1107/s2056989015005642.

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In the title compound, C17H21N5S2, the dihedral angles between the central pyridine ring and its pendant imidazole rings are 29.40 (9) and 40.77 (9)°; the pendant rings are twisted in an opposite sense with respect to the central ring. In each case, the S atom is approximately anti to the N atom of the pyridine ring. For both substituents, the H atom attached to the central C atom of the isopropyl group is approximately syn to the S atom in the attached ring. In the crystal, molecules are linked by weak C—H...S interactions, generatingC(5) chains propagating along [001].

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23

Jespers, Eric. "Special Principal Ideal Rings and Absolute Subretracts." Canadian Mathematical Bulletin 34, no. 3 (September 1, 1991): 364–67. http://dx.doi.org/10.4153/cmb-1991-058-6.

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AbstractA ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

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24

DOBBS, DAVID E., and JAY SHAPIRO. "A GENERALIZATION OF PRÜFER'S ASCENT RESULT TO NORMAL PAIRS OF COMPLEMENTED RINGS." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1351–62. http://dx.doi.org/10.1142/s021949881100521x.

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Let R ⊆ T be a (unital) extension of (commutative) rings, such that the total quotient ring of R is a von Neumann regular ring and T is torsion-free as an R-module. Let T ⊆ B be a ring extension such that B is a reduced ring that is torsion-free as a T-module. Let R* (respectively, A) be the integral closure of R in T (respectively, in B). Then (R*, T) is a normal pair (i.e. S is integrally closed in T for each ring S such that R* ⊆ S ⊆ T) if and only if (A, AT) is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of complemented rings.

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25

Ben, Nasr, and Nabil Zeidi. "A note on the FIP property for extensions of commutative rings." Filomat 33, no. 19 (2019): 6213–18. http://dx.doi.org/10.2298/fil1919213b.

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A ring extension R ? S is said to be FIP if it has only finitely many intermediate rings between R and S. The main purpose of this paper is to characterize the FIP property for a ring extension, where R is not (necessarily) an integral domain and S may not be an integral domain. Precisely, we establish a generalization of the classical Primitive Element Theorem for an arbitrary ring extension. Also, various sufficient and necessary conditions are given for a ring extension to have or not to have FIP, where S = R[?] with ? a nilpotent element of S.

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26

Wu, Huang, Yu Wang, Leighton O. Jones, Wenqi Liu, Bo Song, Yunpeng Cui, Kang Cai, et al. "Ring-in-Ring(s) Complexes Exhibiting Tunable Multicolor Photoluminescence." Journal of the American Chemical Society 142, no. 39 (September 4, 2020): 16849–60. http://dx.doi.org/10.1021/jacs.0c07745.

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27

&NA;. "World??s first estrogen ring." Inpharma Weekly &NA;, no. 882 (April 1993): 23. http://dx.doi.org/10.2165/00128413-199308820-00050.

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28

Sword, Helen. "H. D.'s Majic Ring." Tulsa Studies in Women's Literature 14, no. 2 (1995): 347. http://dx.doi.org/10.2307/463904.

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29

Patel, Hashmukh S., and Vimal C. Patel. "Polyimides containing s-triazine ring." European Polymer Journal 37, no. 11 (November 2001): 2263–71. http://dx.doi.org/10.1016/s0014-3057(01)00107-0.

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30

Peterson, Gary L. "On the structure of an endomorphism near-ring." Proceedings of the Edinburgh Mathematical Society 32, no. 2 (June 1989): 223–29. http://dx.doi.org/10.1017/s0013091500028625.

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If G is an additive (but not necessarily abelian) group and S is a semigroup of endomorphisms of G, the endomorphism near-ring R of G generated by S consists of all the expressions of the form ɛ1s1+…+ɛnsnwhere ɛi=±1 and si∈S for each i. When functions are written on the right, R forms a distributively generated left near-ring under pointwise addition and composition of functions. A basic reference on near-rings which has a substantial treatment of endomorphism near-rings is [6].

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31

Kumar, K. Mahesh, K. R. Roopashree, M. Vinduvahini, O. Kotresh, and H. C. Devarajegowda. "Crystal structure of (6-bromo-2-oxo-2H-chromen-4-yl)methyl morpholine-4-carbodithioate." Acta Crystallographica Section E Crystallographic Communications 71, no. 7 (June 13, 2015): o489—o490. http://dx.doi.org/10.1107/s2056989015011007.

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In the title compound, C15H14BrNO3S2, the 2H-chromene ring system is nearly planar, with a maximum deviation of 0.034 (2) Å, and the morpholine ring adopts a chair conformation. The dihedral angle between best plane through the 2H-chromene ring system and the morpholine ring is 86.32 (9)°. Intramolecular C—H...S hydrogen bonds are observed. In the crystal, inversion-related C—H...S and C—H...O interactions generateR22(10) andR22(8) rings patterns, respectively. In addition, the crystal packing features π–π interactions between fused benzene rings [centroid–centroid distance = 3.7558 (12) Å].

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32

DOOMS, ANN, and PAULA M. VELOSO. "NORMALIZERS AND FREE GROUPS IN THE UNIT GROUP OF AN INTEGRAL SEMIGROUP RING." Journal of Algebra and Its Applications 06, no. 04 (August 2007): 655–69. http://dx.doi.org/10.1142/s0219498807002429.

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In this article, we introduce the normalizer [Formula: see text] of a subset X of a ring R (with identity) in the unit group [Formula: see text] and consider, in particular, the normalizer of the natural basis ±S of the integral semigroup ring ℤ0S of a finite semigroup S. We investigate properties of this normalizer for the class of semigroup rings of inverse semigroups, which contains, for example, matrix rings, in particular, matrix rings over group rings, and partial group rings. We also construct free groups in the unit group of an integral semigroup ring of a Brandt semigroup using a bicyclic unit.

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33

Gutierrez, Jaime, and Carlos Ruiz De Velasco Y Bellas. "Distributive elements in the near-rings of polynomials." Proceedings of the Edinburgh Mathematical Society 32, no. 1 (February 1989): 73–80. http://dx.doi.org/10.1017/s0013091500006921.

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As usual in the theory of polynomial near-rings, we deal with right near-rings. If N = (N, +,·) is a near-ring, the set of distributive elements of N will be denoted by Nd;It is easy to check that, if N is an abelian near-ring (i.e., r + s = s + r, for all r, s∈N), then Nd is a subring of N.

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KOSLER, KARL A. "EXTENDING TORSION RADICALS." Journal of Algebra and Its Applications 07, no. 01 (February 2008): 91–108. http://dx.doi.org/10.1142/s0219498808002680.

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Over an arbitrary ring R, a symmetric radical is shown to be strongly normalizing. For a fully semiprimary Noetherian ring R, a symmetric radical is normalizing if and only if the class of torsion factor rings of R is closed under ring isomorphisms. In case S is a strongly normalizing or normalizing extension ring of R, a symmetric radical for S is constructed as an extension of a symmetric radical for R. Applications address questions concerning the behavior of Krull dimension and linked prime ideals of S.

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35

Jespers, E., and P. Wauters. "Almost Krull rings." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 2 (October 1986): 275–85. http://dx.doi.org/10.1017/s1446788700033681.

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AbstractThe notion of an almost Krull domain is extended to rings satisfying a polynomial identity. Some general structural results are obtained. We also prove that skew polynomial rings R [ X, σ] remain almost Krull if R is an almost Krull ring. Finally, we study when semigroup ring R[S] are almost Krull rings, in the case when the group of quotients of S has the ascending chain condition on cyclic subgroups. An example is included to show that the general (semi-) group ring case is much more difficult to deal with.

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36

Azarang, A., and O. A. S. Karamzadeh. "On Maximal Subrings of Commutative Rings." Algebra Colloquium 19, spec01 (October 31, 2012): 1125–38. http://dx.doi.org/10.1142/s1005386712000909.

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A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2ℵ0, then | Max (R)| ≤ | RgMax (R)|, where RgMax (R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

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37

Bailey, Abigail C., and John A. Beachy. "On reduced rank of triangular matrix rings." Journal of Algebra and Its Applications 14, no. 04 (February 2015): 1550059. http://dx.doi.org/10.1142/s0219498815500590.

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We determine conditions under which a generalized triangular matrix ring has finite reduced rank, in the general torsion-theoretic sense. These are applied to characterize certain orders in Artinian rings, and to show that if each homomorphic image of a ring S has finite reduced rank, then so does the ring of lower triangular matrices over S.

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38

Janiszewski, Jacek. "Effect of Cross Section Size on Ductility and Fragmentation of Copper Ring at High Strain Rate Loading Conditions." Solid State Phenomena 199 (March 2013): 297–302. http://dx.doi.org/10.4028/www.scientific.net/ssp.199.297.

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In the present work, different copper ring samples geometries with a small aspect ratio of 0.5 or 1 were subjected to an experimental study under electromagnetic expanding ring test conditions. The experimental multiple ring tests were performed under similar loading conditions, that is, the applied maximum expansion velocities covered a range from 128 to 147 m/s (7.4 103s-1) for all ring samples geometries, with the exception of rings with a cross section of 1 mm x 0.5 mm. For these rings, the averaged maximum expansion velocity was higher and equal to 195 m/s (1.2 104s-1). The results of experimental investigations revealed a minor influence of the applied cross section sizes on ductility of copper rings, whereas its fragmentation seems to be dependent on a ring cross section area.

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MESYAN, ZACHARY. "THE IDEALS OF AN IDEAL EXTENSION." Journal of Algebra and Its Applications 09, no. 03 (June 2010): 407–31. http://dx.doi.org/10.1142/s0219498810003999.

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Given two unital associative rings R ⊆ S, the ring S is said to be an ideal (or Dorroh) extension of R if S = R ⊕ I, for some ideal I ⊆ S. In this note, we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.

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40

Anderson, D. D., and Victor Camillo. "Annihilator-semigroup rings." Tamkang Journal of Mathematics 34, no. 3 (September 30, 2003): 223–29. http://dx.doi.org/10.5556/j.tkjm.34.2003.313.

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Let $ R $ be a commutative ring with 1. We define $ R $ to be an annihilator-semigroup ring if $ R $ has an annihilator-Semigroup $ S $, that is, $ (S, \cdot) $ is a multiplicative subsemigroup of $ (R, \cdot) $ with the property that for each $ r \in R $ there exists a unique $ s \in S $ with $ 0 : r = 0 : s $. In this paper we investigate annihilator-semigroups and annihilator-semigroup rings.

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41

Abduldaim, Areej M., and Ahmed M. Ajaj. "Alpha - Skew Pi - Armendariz Rings." International Journal of Advanced Research in Engineering 4, no. 1 (March 29, 2018): 17. http://dx.doi.org/10.24178/ijare.2018.4.1.17.

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In this article we introduce a new concept called Alpha-skew Pi-Armendariz rings (Alpha - S Pi - AR)as a generalization of the notion of Alpha-skew Armendariz rings.Another important goal behind studying this class of rings is to employ it in order to design a modern algorithm of an identification scheme according to the evolution of using modern algebra in the applications of the field of cryptography.We investigate general properties of this concept and give examples for illustration. Furthermore, this paperstudy the relationship between this concept and some previous notions related to Alpha-skew Armendariz rings. It clearly presents that every weak Alpha-skew Armendariz ring is Alpha-skew Pi-Armendariz (Alpha-S Pi-AR). Also, thisarticle showsthat the concepts of Alpha-skew Armendariz rings and Alpha-skew Pi- Armendariz rings are equivalent in case R is 2-primal and semiprime ring.Moreover, this paper proves for a semicommutative Alpha-compatible ringR that if R[x;Alpha] is nil-Armendariz, thenR is an Alpha-S Pi-AR. In addition, if R is an Alpha - S Pi -AR, 2-primal and semiprime ring, then N(R[x;Alpha])=N(R)[x;Alpha]. Finally, we look forwardthat Alpha-skew Pi-Armendariz rings (Alpha-S Pi-AR)be more effect (due to their properties) in the field of cryptography than Pi-Armendariz rings, weak Armendariz rings and others.For these properties and characterizations of the introduced concept Alpha-S Pi-AR, we aspire to design a novel algorithm of an identification scheme.

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Okon, James S., and J. Paul Vicknair. "One-Dimensional Monoid Rings with n-Generated Ideals." Canadian Mathematical Bulletin 36, no. 3 (September 1, 1993): 344–50. http://dx.doi.org/10.4153/cmb-1993-047-3.

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AbstractA commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

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43

Feigelstock, Shalom. "The near-ring of generalized affine transformations." Bulletin of the Australian Mathematical Society 32, no. 3 (December 1985): 345–49. http://dx.doi.org/10.1017/s0004972700002446.

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Blackett and Wolfson studied the near-ring Aff (V) consisting of all affine transformations of a vector space V. This notion is generalized here, and the rear-ring Aff (G) consisting of affine-like maps of a nilpotent group G is introduced. The ideal structure, and the multiplication rule for Aff (G) are determined. Finally a near-ring S is introduced which generalized both Aff (G), and Gonshor's abstract affine near-rings. The ideals of S are determined.

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Brown, K. A., and B. A. F. Wehrfritz. "Goldie criteria for some semiprime rings." Glasgow Mathematical Journal 33, no. 3 (September 1991): 297–308. http://dx.doi.org/10.1017/s0017089500008363.

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We principally consider rings R of the form R = S[G], generated as a ring by the subring S of R and the subgroup G of the group of units of R normalizing S. (All our rings have identities except the nilrings.) We wish to deduce that certain semiprime images of R are Goldie rings from ring theoretic information about S and group theoretic information about G. Usually the latter is given in the form that G/N has some solubility or finiteness property, where N is some specified normal subgroup of G contained in S. Note we do not assume that N = G∩S; in particular N = 〈1〉 is always an option.

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45

SAMMAN, MOHAMMAD. "INVERSE SEMIGROUPS AND SEMINEAR-RINGS." Journal of Algebra and Its Applications 08, no. 05 (October 2009): 713–21. http://dx.doi.org/10.1142/s021949880900362x.

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We investigate the relationship between inverse semigroups and a special class of seminear-rings. We show that when an inverse semigroup T generates a d.g. seminear-ring (S,T) then the representation of the seminear-ring (S,T) is faithful. Some other aspects of representation theory of seminear-rings are discussed in connection with the structure of the underlying semigroups.

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46

Murthy, T. N. Sanjeeva, Zeliha Atioğlu, Mehmet Akkurt, C. S. Chidan Kumar, M. K. Veeraiah, Ching Kheng Quah, and B. P. Siddaraju. "Crystal structure and Hirshfeld surface analysis of (2E)-3-(2,4-dichlorophenyl)-1-(2,5-dichlorothiophen-3-yl)prop-2-en-1-one." Acta Crystallographica Section E Crystallographic Communications 74, no. 9 (August 10, 2018): 1201–5. http://dx.doi.org/10.1107/s2056989018010976.

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The molecular structure of the title compound, C13H6Cl4OS, consists of a 2,5-dichlorothiophene ring and a 2,4-dichlorophenyl ring linked via a prop-2-en-1-one spacer. The dihedral angle between the 2,5-dichlorothiophene ring and the 2,4-dichlorophenyl ring is 12.24 (15)°. The molecule has an E configuration about the C=C bond and the carbonyl group is syn with respect to the C=C bond. The molecular conformation is stabilized by intramolecular C—H...Cl contacts, producing S(6) and S(5) ring motifs. In the crystal, the molecules are linked along the a-axis direction through face-to-face π-stacking between the thiophene rings and the benzene rings of the molecules in zigzag sheets lying parallel to the bc plane along the c axis. The intermolecular interactions in the crystal packing were further analysed using Hirshfield surface analysis, which indicates that the most significant contacts are Cl...H/ H...Cl (20.8%), followed by Cl...Cl (18.7%), C...C (11.9%), Cl...S/S...Cl (10.9%), H...H (10.1%), C...H/H...C (9.3%) and O...H/H...O (7.6%).

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ANDERSON, D. D., and JOHN KINTZINGER. "IDEALS IN DIRECT PRODUCTS OF COMMUTATIVE RINGS." Bulletin of the Australian Mathematical Society 77, no. 3 (June 2008): 477–83. http://dx.doi.org/10.1017/s0004972708000415.

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AbstractLet R and S be commutative rings, not necessarily with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of R×S. Unlike the case where R and S have an identity, an ideal (or primary ideal, or maximal ideal) of R×S need not be a ‘subproduct’ I×J of ideals. We show that for a ring R, for each commutative ring S every ideal (or primary ideal, or maximal ideal) is a subproduct if and only if R is an e-ring (that is, for r∈R, there exists er∈R with err=r) (or u-ring (that is, for each proper ideal A of R, $\sqrt {A}\not =R$)), the Abelian group (R/R2 ,+) has no maximal subgroups).

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48

Sahandi, Parviz, and Siamak Yassemi. "Filter Rings Under Flat Base Change." Algebra Colloquium 15, no. 03 (September 2008): 463–70. http://dx.doi.org/10.1142/s1005386708000448.

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Let φ: (R, 𝔪) → (S, 𝔫) be a flat local homomorphism of rings. In this paper, we prove: (1) If dim S/𝔪S > 0, then S is a filter ring if and only if R and k(𝔭) ⊗R𝔭 S𝔮 are Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭= 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. (2) If dim S/𝔪S = 0, then S is a filter ring if and only if R is a filter ring and k(𝔭) ⊗R𝔭 S𝔮 is Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭 = 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. As an application, it is shown that for a k-algebra R and an algebraic field extension K of k, if K ⊗k R is locally equidimensional, then R is a locally filter ring if and only if K ⊗k R is a locally filter ring.

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Chen, Huanyin, and Marjan Sheibani. "Structure of Zhou Nil-clean Rings." Algebra Colloquium 25, no. 03 (August 14, 2018): 361–68. http://dx.doi.org/10.1142/s1005386718000251.

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A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ϵ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM(R) is 5-potent and BM(R) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn(R; s) is Zhou nil-clean if and only if R is Zhou nil-clean and s ϵ J(R).

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Sevim, Esra Sengelen, Unsal Tekir, and Suat Koc. "S-Artinian rings and finitely S-cogenerated rings." Journal of Algebra and Its Applications 19, no. 03 (March 21, 2019): 2050051. http://dx.doi.org/10.1142/s0219498820500516.

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Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be a multiplicatively closed subset. In this paper, we study [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings. A commutative ring [Formula: see text] is said to be an [Formula: see text]-Artinian ring if for each descending chain of ideals [Formula: see text] of [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text] Also, [Formula: see text] is called a finitely [Formula: see text]-cogenerated ring if for each family of ideals [Formula: see text] of [Formula: see text] where [Formula: see text] is an index set, [Formula: see text] implies [Formula: see text] for some [Formula: see text] and a finite subset [Formula: see text] Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings.

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